Polynomial taylor

Question

Suppose P2(x)= A+Bx+C(x^2) is the second-degree Taylor polynomial for the function f(x) about x=0 shown in the figure below. Fill in the information below about the coefficients (A,B,C) of p2(x), Justify your answers with a sentence each. The figure given has a concave down graph of p2(x) which goes through the points (-2,-3),(0,-1), and (2,-1). The figure also has a graph of f(x) that goes through (-2,-2.8), (0,-1) and (2,-0.5). (a) A is positive, negative or zero? (b) B is positive, negative or zero? (c) C is positive negative or zero?

Answer 1

P₂(x) = f(0) + f'(0)x + [f"(0)/2]x² = A + Bx + Cx²

So, A = f(0) = -1 < 0

The graph of P₂(x) is a parabola. Since the graph is concave down, C < 0.

Check the given graph of f(x) to see if the graph is increasing, decreasing, or has a horizontal tangent when x = 0. If increasing, then B = f'(0) > 0. If decreasing, then B = f'(0) < 0. If there is a horizontal tangent when x = 0, then B = f'(0) = 0.